On Inferences (1950-1974)

"The study of inductive inference belongs to the theory of probability, since observational facts can make a theory only probable but will never make it absolutely certain." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"The technical analysis of any large collection of data is a task for a highly trained and expensive man who knows the mathematical theory of statistics inside and out. Otherwise the outcome is likely to be a collection of drawings - quartered pies, cute little battleships, and tapering rows of sturdy soldiers in diversified uniforms - interesting enough in the colored Sunday supplement, but hardly the sort of thing from which to draw reliable inferences." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)

"Statistics is the name for that science and art which deals with uncertain inferences - which uses numbers to find out something about nature and experience." (Warren Weaver, 1952)

"The heart of all major discoveries in the physical sciences is the discovery of novel methods of representation, and so of fresh techniques by which inferences can be drawn - and drawn in ways which fit the phenomena under investigation." (Stephen Toulmin, "The Philosophy of Science: An introduction", 1953)

"From the outset it was clear that the two kinds of reasoning have different tasks. From the outset. they appeared very different: demonstrative reasoning as definite, final, 'machinelike'; and plausible reasoning as vague, provisional, specifically 'human'. Now we may see the difference a little more distinctly. In opposition to demonstrative inference, plausible inference leaves indeterminate a highly relevant point: the 'strength' or the 'weight' of the conclusion. This weight may depend not only on clarified grounds such as those expressed in the premises, hut also on unclarified unexpressed grounds somewhere on the background of the person who draws the conclusion. A person has a background, a machine has not. Indeed, you can build a machine to draw demonstrative conclusions for you, but I think you can never build a machine that will draw plausible inferences." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya, "Induction and Analogy in Mathematics", 1954)

"The heart of all major discoveries in the physical sciences is the discovery of novel methods of representation and so of fresh techniques by which inferences can be drawn - and drawn in ways which fit the phenomena under investigation." (Stephen Toulmin, "The Philosophy of Science", 1957)

"The first [principle], is that a mathematical theory can only he developed axiomatically in a fruitful way when the student has already acquired some familiarity with the corresponding material - a familiarity gained by working long enough with it on a kind of experimental, or semiexperimental basis, i.e. with constant appeal to intuition. The other principle [...]  is that when logical inference is introduced in some mathematical question, it should always he presented with absolute honesty - that is, without trying to hide gaps or flaws in the argument; any other way, in my opinion, is worse than giving no proof at all." (Jean Dieudonné, "Thinking in School Mathematics", 1961)

"Statistics is that branch of mathematics which deals with the accumulation and analysis of quantitative data. There are three principal subdivisions in the field of statistics but these overlap, more often than not, in actual practice. First, inference from samples to population by means of probability is called statistical inference. Second, descriptive statistics is defined as the characterization and summarization of a given set of data without direct reference to inference. And finally, sampling statistics deals with methods of obtaining samples for statistical inference." (David B MacNeil, "Modern Mathematics for the Practical Man", 1963)

"A mathematical proof, as usually written down, is a sequence of expressions in the state space. But we may also think of the proof as consisting of the sequence of justifications of consecutive proof steps - i.e., the references to axioms, previously-proved theorems, and rules of inference that legitimize the writing down of the proof steps. From this point of view, the proof is a sequence of actions (applications of rules of inference) that, operating initially on the axioms, transform them into the desired theorem." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

"Inductive inference is the only process known to us by which essential new knowledge comes into the world." (Sir Ronald A Fisher, "The Design of Experiments", 1971)

"Probability theory, for us, is not so much a part of mathematics as a part of logic, inductive logic, really. It provides a consistent framework for reasoning about statements whose correctness or incorrectness cannot be deduced from the hypothesis. The information available is sufficient only to make the inferences 'plausible' to a greater or lesser extent.". (Ralph Baierlein, "Atoms and Information Theory: An Introduction to Statistical Mechanics", 1971)

"The statistician cannot excuse himself from the duty of getting his head clear on the principles of scientific inference, but equally no other thinking man can avoid a like obligation." (Sir Ronald A Fisher, "The Design of Experiments", 1971)

"[...] we will adopt the broad view and will take 'probability', to be a quantitative relation, between a hypothesis and an inference, corresponding to the degree of rational belief in the correctness of the inference, given the hypothesis. The hypothesis is the information we possess, or assume for the sake of argument. The inference is a statement that, to a greater or lesser extent, is justified by the hypothesis. Thus 'the probability' of an inference, given a hypothesis, is the degree of rational belief in the correctness of the inference, given the hypothesis." (Ralph Baierlein, "Atoms and Information Theory: An Introduction to Statistical Mechanics", 1971)

"An analogy is a relationship between two entities, processes, or what you will, which allows inferences to be made about one of the things, usually that about which we know least, on the basis of what we know about the other. […] The art of using analogy is to balance up what we know of the likenesses against the unlikenesses between two things, and then on the basis of this balance make an inference as to what is called the neutral analogy, that about which we do not know." (Rom Harré," The Philosophies of Science", 1972)

"The process [of statistical analysis] usually begins by the postulating of a model worthy to be tentatively entertained. The data analyst will have arrived at this tentative model in cooperation with the scientific investigator. They will choose it 'So that, in the light of the then available knowledge, it best takes account of relevant phenomena in the simplest way possible. it will usually contain unknown parameters. Given the data the analyst can now make statistical inferences about the parameters conditional on the correctness of this first tentative model. These inferences form part of the conditional analysis. If the model is correct, they provide all there is to know about the problem under study, given the data." (George E P Box & George C Tjao, "Bayesian Inference in Statistical Analysis", 1973)

"[…] it is not enough to say: 'There's error in the data and therefore the study must be terribly dubious'. A good critic and data analyst must do more: he or she must also show how the error in the measurement or the analysis affects the inferences made on the basis of that data and analysis." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

On Inferences (1900-1949)

"I may as well say at once that I do not distinguish between inference and deduction. What is called induction appears to me to be either disguised deduction or a mere method of making plausible guesses." (Bertrand Russell, "Principles of Mathematics", 1903)

"A theorem […] is an inference obtained by constructing a diagram according to a general precept, and after modifying it as ingenuity may dictate, observing in it certain relations, and showing that they must subsist in every case, retranslating the proposition into general terms." (Charles S Peirce, "New Elements" ["Kaina stoiceia"], 1904)

"The type of reasoning found in mathematics seems thus not only available but essentially interwoven with every inference in non-mathematical reasoning, being always used in one of its two steps ; facility in making the other step, the more difficult one, must be attained through other than purely mathematical training." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"It is experience which has given us our first real knowledge of Nature and her laws. It is experience, in the shape of observation and experiment, which has given us the raw material out of which hypothesis and inference have slowly elaborated that richer conception of the material world which constitutes perhaps the chief, and certainly the most characteristic, glory of the modern mind." (Arthur J Balfour, "The Foundations of Belief", 1912)

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"Whenever possible, substitute constructions out of known entities for inferences to unknown entities." (Bertrand Russell, 1924)

"Hypothesis, however, is an inference based on knowledge which is insufficient to prove its high probability." (Frederick L Barry, "The Scientific Habit of Thought", 1927) 

"The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. [...] One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms." (Kurt Gödel, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems", 1931)

"An inference, if it is to have scientific value, must constitute a prediction concerning future data. If the inference is to be made purely with the help of the distribution theory of statistics, the experiments that constitute evidence for the inference must arise from a state of statistical control; until that state is reached, there is no universe, normal or otherwise, and the statistician’s calculations by themselves are an illusion if not a delusion. The fact is that when distribution theory is not applicable for lack of control, any inference, statistical or otherwise, is little better than a conjecture. The state of statistical control is therefore the goal of all experimentation." (William E Deming, "Statistical Method from the Viewpoint of Quality Control", 1939)

"An observation, strictly, is only a sensation. Nobody means that we should reject everything but sensations. But as soon as we go beyond sensations we are making inferences." (Sir Harold Jeffreys, "Theory of Probability", 1939)

"Inference by analogy appears to be the most common kind of conclusion, and it is possibly the most essential kind. It yields more or less plausible conjectures which may or may not be confirmed by experience and stricter reasoning." (George Pólya, "How to Solve It", 1945)

"If the chance of error alone were the sole basis for evaluating methods of inference, we would never reach a decision, but would merely keep increasing the sample size indefinitely." (C West Churchman, "Theory of Experimental Inference", 1948)

On Inferences (-1899)

"Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." (Eudoxus, cca. 4th century BC)

"Every stage of science has its train of practical applications and systematic inferences, arising both from the demands of convenience and curiosity, and from the pleasure which, as we have already said, ingenious and active-minded men feel in exercising the process of deduction." (William Whewell, "The Philosophy of the Inductive Sciences Founded Upon Their History", 1840)

"There is in every step of an arithmetical or algebraical calculation a real induction, a real inference from facts to facts, and what disguises the induction is simply its comprehensive nature, and the consequent extreme generality of its language." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)

"A mere inference or theory must give way to a truth revealed; but a scientific truth must be maintained, however contradictory it may appear to the most cherished doctrines of religion." (David Brewster, "More Worlds Than One: The Creed of the Philosopher and the Hope of the Christian", 1856)

"Truths are known to us in two ways: some are known directly, and of themselves; some through the medium of other truths. The former are the subject of Intuition, or Consciousness; the latter, of Inference; the latter of Inference. The truths known by Intuition are the original premises, from which all others are inferred." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1858)

"And first, it is necessary to distinguish from true inductions, certain operations which are often improperly called by that name. A true induction is a process of inference - it proceeds from the known to the unknown; and whenever any operation contains no inference, it is not  really an induction. And yet most of the examples given in the common  works on logic, as examples of induction, are of this character." (George R Drysdale, "Logic and Utility: The tests of truth and falsehood, and of right and wrong", 1866)

"It must be the ground of all reasoning and inference that what is true of one thing will be true of its equivalent, and that under carefully ascertained conditions Nature repeats herself." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Economic science is but the working of common sense aided by appliances of organized analysis and general reasoning, which facilitate the task of collecting, arranging, and drawing inferences from particular facts. Though its scope is always limited, though its work without the aid of common sense is vain, yet it enables common sense to go further in difficult problems than would otherwise be possible." (Alfred Marshall, "Principles of Economics", 1890)

Experience - Trivia

"It is frequently analogy which guides the experienced to what are called good guesses." (Francis W Newman, "Lectures on Logic", 1838)

"An experiment is an observation that can be repeated, isolated and varied. The more frequently you can repeat an observation, the more likely are you to see clearly what is there and to describe accurately what you have seen. The more strictly you can isolate an observation, the easier does your task of observation become, and the less danger is there of your being led astray by irrelevant circumstances, or of placing emphasis on the wrong point. The more widely you can vary an observation, the more clearly will be the uniformity of experience stand out, and the better is your chance of discovering laws." (Edward B Titchener, "A Text-Book of Psychology", 1909)

"Our system of philosophy is itself on trial; it must stand or fall according as it is broad enough to find room for this experience as an element of life." (Sir Arthur S Eddington, "Science and the Unseen World", 1929)

"One has to recognize that science is not metaphysics, and certainly not mysticism; it can never bring us the illumination and the satisfaction experienced by one enraptured in ecstasy. Science is sobriety and clarity of conception, not intoxicated vision." (Ludwig Von Mises, "Epistemological Problems of Economics", 1933)

"The great extension of our experience in recent years has brought light to the insufficiency of our simple mechanical conceptions and, as a consequence, has shaken the foundation on which the customary interpretation of observation was based." (Niels Bohr, "Atomic Physics and the Description of Nature", 1934)

"[T]he sudden inventions characteristic of the sixth stage [of infant development] are in reality the product of a long evolution of schemata and not only of an internal maturation of perceptive structures. [..] This is revealed by the existence of a fifth stage, characterized by experimental groping. […] What does this mean if not that the practice of actual experience is necessary in order to acquire the practice of mental experience and that invention does not arise entirely preformed despite appearances? (Jean Piaget, "The origin of intelligence in children" 1936)

"We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but if such a problem could exist, it would be insoluble. In fact, when solving a problem, we should always profit from previously solved problems, using their result or their method, or the experience acquired in solving them." (George Polya, 1945)

"We have here no esoteric theory of the ultimate nature of concepts, nor a philosophical championing of the primacy of the 'operation'. We have merely a pragmatic matter, namely that we have observed after much experience that if we want to do certain kinds of things with our concepts, our concepts had better be constructed in certain ways. In fact one can see that the situation here is no different from what we always find when we push our analysis to the limit; operations are not ultimately sharp or irreducible any more than any other sort of creature. We always run into a haze eventually, and all our concepts are describable only in spiralling approximation." (Percy W Bridgman, "Reflections of a Physicist", 1950)

"Modern scientific principle has been drawn from the investigation of natural laws, technology has developed from the experience of doing, and the two have been combined by means of mathematical system to form what we call engineering." (George S Emmerson, "Engineering Education: A Social History", 1973)

"Models are not intended to either reflect or construct a single objective reality. Rather, their purpose is to simulate some aspect of a possible reality. In NLP, for instance, it is not important whether or not a model is 'true' , but rather that it is 'useful' . In fact, all models can be perceived as symbolic or metaphoric, as opposed to reflective of reality. Whether the description being used is metaphorical or literal, the usefulness of a model depends on the degree to which it allows us to move effectively to the next step in the sequence of transformations connecting deeper structures and surface structures. Instead of 'constructing' reality, models establish a set of functions that serve as a tool or a bridge between deep structures and surface structures. It is this bridge that forms our 'understanding' of reality and allows us to generate new experiences and expressions of reality." (Richard Bandler & John Grinder, "The Structure of Magic", 1975)

"It is often the scientist’s experience that he senses the nearness of truth when such connections are envisioned. A connection is a step toward simplification, unification. Simplicity is indeed often the sign of truth and a criterion of beauty." (Mahlon B Hoagland, "Toward the Habit of Truth", 1990)

"Systems theory pursues the scientific exploration and understanding of systems that exist in the various realms of experience, in order to arrive at a general theory of systems: an organized expressing of sets of interrelated concepts and principles that apply to all systems." (Béla H Bánáthy, "Systems Design of Education", 1991)

"It is in the nature of exponential growth that events develop extremely slowly for extremely long periods of time, but as one glides through the knee of the curve, events erupt at an increasingly furious pace. And that is what we will experience as we enter the twenty-first century." (Ray Kurzweil, "The Age of Spiritual Machines: When Computers Exceed Human Intelligence", 1999)

"Cultural archetypes are the unconscious models that help us make sense of the world: they are the myths, narratives, images, symbols, and files into which we organize the data of our life experience" (Clotaire Rapaille, "Cultural Imprints", Executive Excellence Vol. 16 (10), 1999)

"Learning is a process of modifying or completely changing our mental models based on new experiences or evidence." (Edward D Hess, "Learn or Die: Using Science to Build a Leading-Edge Learning Organization", 2014)

"What is consciousness? Our brain simulates reality. So, our everyday experiences are a form of dreaming, which is to say, they are mental models, simulations, not the things they appear to be." (Stephen LaBerge, "Losi in Lucidity", 2014)

John McCarthy - Collected Quotes

"[…] the difference between creative thinking and unimaginative competent thinking lies in the injection of a some randomness. The randomness must be guided by intuition to be efficient." (John McCarthy et al, "A Proposal for the Dartmouth Summer Research Project on Artificial Intelligence", 1955)

"The following are some aspects of the artificial intelligence problem: […] If a machine can do a job, then an automatic calculator can be programmed to simulate the machine. […] It may be speculated that a large part of human thought consists of manipulating words according to rules of reasoning and rules of conjecture. From this point of view, forming a generalization consists of admitting a new word and some rules whereby sentences containing it imply and are implied by others. This idea has never been very precisely formulated nor have examples been worked out. […] How can a set of (hypothetical) neurons be arranged so as to form concepts. […] to get a measure of the efficiency of a calculation it is necessary to have on hand a method of measuring the complexity of calculating devices which in turn can be done. […] Probably a truly intelligent machine will carry out activities which may best be described as self-improvement. […] A number of types of 'abstraction' can be distinctly defined and several others less distinctly. […] the difference between creative thinking and unimaginative competent thinking lies in the injection of a some randomness. The randomness must be guided by intuition to be efficient." (John McCarthy et al, "A Proposal for the Dartmouth Summer Research Project on Artificial Intelligence", 1955)

"We call a problem well-defined if there is a test which can be applied to a proposed solution. In case the proposed solution is a solution, the test must confirm this in a finite number of steps." (John McCarthy, "The Inversion of Functions Denned by Turing Machines", 1956)

"We shall therefore say that a program has common sense if it automatically deduces for itself a sufficient wide class of immediate consequences of anything it is told and what it already knows. [...] Our ultimate objective is to make programs that learn from their experience as effectively as humans do. We shall [...] say that a program has common sense if it automatically deduces for itself a sufficient wide class of immediate consequences of anything it is told and what it already knows" (John McCarthy, "Programs with Common Sense", 1958)

"Intelligence has two parts, which we shall call the epistemological and the heuristic. The epistemological part is the representation of the world in such a form that the solution of problems follows from the facts expressed in the representation. The heuristic part is the mechanism that on the basis of the information solves the problem and decides what to do." (John McCarthy & Patrick J Hayes, "Some Philosophical Problems from the Standpoint of Artificial Intelligence", Machine Intelligence 4, 1969)

"The right way to think about the general problems of metaphysics and epistemology is not to attempt to clear one's own mind of all knowledge and start with 'Cogito ergo sum' and build up from there. Instead, we propose to use all of our knowledge to construct a computer program that knows. The correctness of our philosophical system will be tested by numerous comparisons between the beliefs of the program and our own observations and knowledge." (John McCarthy & Patrick J. Hayes, "Some Philosophical Problems from the Standpoint of Artificial Intelligence", 1969)

"[This] is or should be our main scientific activity - studying the structure of information and the structure of problem-solving processes independently of applications and independently of its realization in animals or humans." (John McCarthy, 1974)

"When we program a computer to make choices intelligently after determining its options, examining their consequences, and deciding which is most favorable or most moral or whatever, we must program it to take an attitude towards its freedom of choice essentially isomorphic to that which a human must take to his own." (John McCarthy, "Ascribing Mental Qualities to Machines", 1979) 

"It's difficult to be rigorous about whether a machine really 'knows', 'thinks', etc., because we're hard put to define these things. We understand human mental processes only slightly better than a fish understands swimming." (John McCarthy, "The Little Thoughts of Thinking Machines", Psychology Today, 1983)

"Program designers have a tendency to think of the users as idiots who need to be controlled. They should rather think of their program as a servant, whose master, the user, should be able to control it. If designers and programmers think about the apparent mental qualities that their programs will have, they'll create programs that are easier and pleasanter - more humane - to deal with." (John McCarthy, "The Little Thoughts of Thinking Machines", Psychology Today,1983)

"Whenever we write an axiom, a critic can say that the axiom is true only in a certain context. With a little ingenuity the critic can usually devise a more general context in which the precise form of the axiom doesn't hold. [...] There simply isn't a most general context." (John McCarthy, "Generality in Artificial Intelligence", 1987)

"I don't see that human intelligence is something that humans can never understand." (John McCarthy, 1989)

On Standards II: Mathematics II

"Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics." (Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly Vol. 87 (2), 1980)

"Thus statistics should generally be taught more as a practical subject with analyses of real data. Of course some theory and an appropriate range of statistical tools need to be learnt, but students should be taught that Statistics is much more than a collection of standard prescriptions." (Christopher Chatfield, "The Initial Examination of Data", Journal of the Royal Statistical Society A Vol. 148, 1985)

"[…] calling upon the needs of rigor to explain the development of mathematics constitutes a circular argument. In actual fact, new standards of rigor are formed when the old criteria no longer permit an adequate response to questions that arise in mathematical practice or to problems that are in a certain sense external to mathematics. When these are treated mathematically, they compel changes in the theoretical framework of mathematics. It is thus not by chance that mathematical physics and applied mathematics have generally been formidable stimuli to the development of pure mathematics." (Umberto Bottazzini, "The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass", 1986)

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth." (National Research Council, "Everybody Counts", 1989)

"'Technique' is a term used equally by chess players and mathematicians to describe sequences of moves which are standard, familiar and unoriginal. Once upon a time, the particular technique was an invention, a new discovery, but no longer. The precise sequence of moves required may never have been played before in the history of the world, yet no new ideas, no originality and no imagination are demanded, at least of the experienced player. (To the learner, of course, the most mundane sequences will appear novel and require original thought.)" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Elegance and simplicity should remain important criteria in judging mathematics, but the applicability and consequences of a result are also important, and sometimes these criteria conflict. I believe that some fundamental theorems do not admit simple elegant treatments, and the proofs of such theorems may of necessity be long and complicated. Our standards of rigor and beauty must be sufficiently broad and realistic to allow us to accept and appreciate such results and their proofs. As mathematicians we will inevitably use such theorems when it is necessary in the practice our trade; our philosophy and aesthetics should reflect this reality." (Michael Aschbacher,"Highly complex proofs and implications", 2005)

“When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze.” (William P Thurston, [preface to (John H Hubbard, “Teichmüller Theory and Applications”,Vol. 1, 2006)])

"Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress." (Edward Frenkel, "Love and Math", 2014)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

On Standards I: Mathematics I

"Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case we can only determine the more or less and not the how much." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers)." (Henry B Fine, "The Number-System of Algebra", 1890)

"The main source of mathematical invention seems to be within man rather than outside of him: his own inveterate and insatiable curiosity, his constant itching for intellectual adventure; and likewise the main obstacles to mathematical progress seem to be also within himself; his scandalous inertia and laziness, his fear of adventure, his need of conformity to old standards, and his obsession by mathematical ghosts." (George Sarton, "The Study of the History of Mathematics", 1936)

"Mathematicians themselves set up standards of generality and elegance in their exposition which are a bar to understand." (Kenneth E Boulding, "Economic Analysis", 1941)

"As an Art, Mathematics has its own standard of beauty and elegance which can vie with the more decorative arts. In this it is diametrically opposed to a Baroque art which relies on a wealth of ornamental additions. Bereft of superfluous addenda, Mathematics may appear, on first acquaintance, austere and severe. But longer contemplation reveals the classic attributes of simplicity relative to its significance and depth of meaning." (Dudley E Littlewood,"The Skeleton Key of Mathematics", 1949)

"Demonstrative reasoning is safe, beyond controversy, and final. Plausible reasoning is hazardous, controversial, and provisional. Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning, for which we care in everyday affairs. Demonstrative reasoning has rigid standards, codified and clarified by logic (formal or demonstrative logic), which is the theory of demonstrative reasoning. The standards of plausible reasoning are fluid, and there is no theory of such reasoning that could be compared to demonstrative logic in clarity or would command comparable consensus." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better." (Paul Dirac, "The Evolution of the Physicist's Picture of Nature", 1963)

"When we propose to apply mathematics we are stepping outside our own realm, and such a venture is not without dangers. For having stepped out, we must be prepared to be judged by standards not of our own making and to play games whose rules have been laid down with little or no consultation with us. Of course, we do not have to play, but if we do we have to abide by the rules and above all not try to change them merely because we find them uncomfortable or restrictive." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972)

"[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility." (Lynn A Steen, "Mathematics Today: Twelve Informal Essays", Mathematics Today, 1978)

Out of Context: On Thought (Definitions)

"It is probable that what we call thought is not an actual being, but no more than the relation between certain parts of that infinitely varied mass, of which the rest of the universe is composed, and which ceases to exist as soon as those parts change their position with regard to each other." (Percy B Shelley, "On a Future State", 1815)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises." (George H Lewes "Problems of Life and Mind", 1873)

"[...] thought is the representative or cognitive apprehension of relations among notions; imagination is the affective or felt apprehension of relations among images." (James M Baldwin,"Handbook of Psychology: Senses and Intellect", 1890)

"Thought is existence. More than that, so far as we are concerned, existence is thought, all our conceptions of existence being some kind or other of thought." (Thomas H Huxley, "Method and Results", 1893)

"Consequently, all truly strict and exact thought is sustained by the symbolic and semiotics on which it is based." (Ernst Cassirer, "The Philosophy of Symbolic Forms", 1923)

"Thought is prior to language and consists in the simultaneous presentation to the mind of two different images." (Thomas E Hulme, "Notes on Language and Style", 1929)

"Analytic thought is based on detailed defined relations between two elements at a time. Intuitive thought is based on an emotional state associated with all the elements in the field of knowledge (overall impression). " (Tony Bastick, "Intuition: How we think and act", 1982) 

Mathematicians vs Scientists

"The man of science, who, forgetting the limits of philosophical inquiry, slides from these formulæ and symbols into what is commonly understood by materialism, seems to me to place himself on a level with the mathematician, who should mistake the x's and y's with which he works his problems for real entities - and with this further disadvantage, as compared with the mathematician, that the blunders of the latter are of no practical consequence, while the errors of systematic materialism may paralyse the energies and destroy the beauty of a life." (Thomas H Huxley, "Method and Results", 1893)

"Mathematicians and other scientists, however great they may be, do not know the future. Their genius may enable them to project their purpose ahead of them; it is as if they had a special lamp, unavailable to lesser men, illuminating their path; but even in the most favorable cases the lamp sends only a very small cone of light into the infinite darkness." (George Sarton, "The Study of the History of Mathematics", 1936)

"One of the difficulties which a mathematician has in describing his work to non-mathematicians is that the present day language of mathematics has become so esoteric that a well educated layman, or even a group of scientists, can comprehend essentially nothing of the discourse which mathematicians hold with each other, or of the accounts of their latest researches which are published in their professional journals." (Angus E Taylor," Some Aspects of Mathematical Research", American Scientist , Vol. 35, No. 2, 1947)

"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature. [...] we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it." (Henri Poincaré, "Notice sur Halphen", Journal de l'École Polytechnique, 1890)

"The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it - to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions." (Ian Stewart, "Nature's Numbers", 1995)

"Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought." (Clifford A Pickover, "The Math Book", 2009)

"The reasoning of the mathematician and that of the scientist are similar to a point. Both make conjectures often prompted by particular observations. Both advance tentative generalizations and look for supporting evidence of their validity. Both consider specific implications of their generalizations and put those implications to the test. Both attempt to understand their generalizations in the sense of finding explanations for them in terms of concepts with which they are already familiar. Both notice fragmentary regularities and - through a process that may include false starts and blind alleys - attempt to put the scattered details together into what appears to be a meaningful whole. At some point, however, the mathematician’s quest and that of the scientist diverge. For scientists, observation is the highest authority, whereas what mathematicians seek ultimately for their conjectures is deductive proof." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures and Proofs", 2009)

💠On Problem Solving 16 (Trivia)

"Paradoxes have played a dramatic part in intellectual history, often foreshadowing revolutionary developments in science, mathematics, and logic. Whenever, in any discipline, we discover a problem that cannot be solved within the conceptual framework that supposedly should apply, we experience an intellectual shock. The shock may compel us to discard the old framework and adopt a new one. It is to this process of intellectual molting that we owe the birth of many of the major ideas in mathematics and science." (Anatol Rapoport, "Escape from Paradox", Scientific American Vol. 217 (1), 1967)

"The problems of different fields are much more alike than their practitioners think, much more alike than different." (John W Tukey, "Analyzing Data: Sanctification or Detective Work?", 1969)

"Design is really a special case of problem solving. One wants to bring about a desired state of affairs. Occasionally one wants to remedy some fault but more usually one wants to bring about something new. For that reason design is more open ended than problem solving. It requires more creativity. It is not so much a matter of linking up a clearly defined objective with a clearly defined starting position (as in problem solving) but more a matter of starting out from a general position in the direction of a general objective." (Edward de Bono, "Lateral Thinking: Creativity Step by Step", 1970)

"A system represents someone's solution to a problem. The system doesn't solve the problem." (John Gall, "General Systemantics: How systems work, and especially how they fail", 1975)

"[...] any hope that we are smart enough to find even transiently optimum solutions to our data analysis problems is doomed to failure, and, indeed, if taken seriously, will mislead us in the allocation of effort, thus wasting both intellectual and computational effort." (John W Tukey, "Choosing Techniques for the Analysis of Data", 1981)

"We will treat problem solving as a process of search through a state space. A problem is defined by an initial state, one or more goal states to be reached, a set of operators that can transform one state into another, and constraints that an acceptable solution must meet. Problem-solving methods are procedures for selecting an appropriate sequence of operators that will succeed in transforming the initial state into a goal state through a series of steps." (John H Holland et al, "Induction: Processes Of Inference, Learning, And Discovery", 1986)

"Delay time, the time between causes and their impacts, can highly influence systems. Yet the concept of delayed effect is often missed in our impatient society, and when it is recognized, it’s almost always underestimated. Such oversight and devaluation can lead to poor decision making as well as poor problem solving, for decisions often have consequences that don’t show up until years later. Fortunately, mind mapping, fishbone diagrams, and creativity/brainstorming tools can be quite useful here." (Stephen G Haines, "The Managers Pocket Guide to Systems Thinking & Learning", 1998)

"A problem is defined as a gap between where you are and where you want to be that is confronted with obstacles that make closing the gap difficult. It is actually the obstacles that make the gap a problem." (James P Lewis, "Project Planning, Scheduling, and Control" 3rd Ed., 2001)

"Complex problems are the problems that persist - the problems that bounce back and continue to haunt us. People often go through a series of stages in dealing with such problems - from believing they are beyond hope, to galvanizing collective efforts of many people and dollars to address the problem, to despair, retreat, and rationalization."(Yaneer Bar-Yam, "Making Things Work: Solving Complex Problems in a Complex World", 2004)

"Design thinking taps into capacities we all have but that are overlooked by more conventional problem-solving practices. It is not only human-centered; it is deeply human in and of itself. Design thinking relies on our ability to be intuitive, to recognize patterns, to construct ideas that have emotional meaning as well as functionality, to express ourselves in media other than words or symbols." (Tim Brown, "Change by Design: How Design Thinking Transforms Organizations and Inspires Innovation", 2009)

"Mostly we rely on stories to put our ideas into context and give them meaning. It should be no surprise, then, that the human capacity for storytelling plays an important role in the intrinsically human-centered approach to problem solving, design thinking." (Tim Brown, "Change by Design: How Design Thinking Transforms Organizations and Inspires Innovation", 2009)

"The basic idea behind all of these techniques is to simplify problem solving by concentrating on its essentials. Consolidate and simplify the objectives. Focus on the things with the highest impact, things that determine other things. Put to one side minor issues likely to be resolved by the resolution of major ones. Discard the nonessentials. Model (abstract) the system at as high a level as possible, then progressively reduce the level of abstraction. In short: Simplify!" (Mark W Maier, "The Art Systems of Architecting" 3rd Ed., 2009)

"Teaching statistical thinking means giving people tools for problem solving in the real world. It should not be taught as pure mathematics. Instead of mechanically solving a dozen problems with the help of a particular formula, children and adolescents should be asked to find solutions to real-life problems. That’s what teaches them how to solve problems, and also shows that there may be more than one good answer in the first place. Equally important is encouraging curiosity, such as asking for answers to questions by doing experiments." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"Building a comprehensive problem-solving framework is about leveraging a structured methodology that allows you to frame problems systematically and solve problems creatively." (Pearl Zhu, "Problem Solving Master: Frame Problems Systematically and Solve Problem Creatively", 2017)

"[...] strategy is about determining the problems and opportunities in front of you, defining them properly, and shaping a course of action that will give your business the greatest advantage. Balancing problem solving with creating and exploiting new opportunities through imagination and analysis is the cornerstone of a great strategy." (Eben Hewitt, "Technology Strategy Patterns: Architecture as strategy" 2nd Ed., 2019)

"The most fruitful research grows out of practical problems."  (Ralph B Peck)

"Understand the reality of the problem first, rather than wondering what its cause may be. The first step of problem solving is to understand the existing conditions." (Kaoru Ishikawa)

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